UnitBezier.h   [plain text]

/*
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
*    notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
*    notice, this list of conditions and the following disclaimer in the
*    documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY APPLE INC. ``AS IS'' AND ANY
* EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL APPLE INC. OR
* CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
* EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
* PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
* PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
* OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/

#ifndef UnitBezier_h
#define UnitBezier_h

#include <math.h>

namespace WebCore {

struct UnitBezier {
UnitBezier(double p1x, double p1y, double p2x, double p2y)
{
// Calculate the polynomial coefficients, implicit first and last control points are (0,0) and (1,1).
cx = 3.0 * p1x;
bx = 3.0 * (p2x - p1x) - cx;
ax = 1.0 - cx -bx;

cy = 3.0 * p1y;
by = 3.0 * (p2y - p1y) - cy;
ay = 1.0 - cy - by;
}

double sampleCurveX(double t)
{
// `ax t^3 + bx t^2 + cx t' expanded using Horner's rule.
return ((ax * t + bx) * t + cx) * t;
}

double sampleCurveY(double t)
{
return ((ay * t + by) * t + cy) * t;
}

double sampleCurveDerivativeX(double t)
{
return (3.0 * ax * t + 2.0 * bx) * t + cx;
}

// Given an x value, find a parametric value it came from.
double solveCurveX(double x, double epsilon)
{
double t0;
double t1;
double t2;
double x2;
double d2;
int i;

// First try a few iterations of Newton's method -- normally very fast.
for (t2 = x, i = 0; i < 8; i++) {
x2 = sampleCurveX(t2) - x;
if (fabs (x2) < epsilon)
return t2;
d2 = sampleCurveDerivativeX(t2);
if (fabs(d2) < 1e-6)
break;
t2 = t2 - x2 / d2;
}

// Fall back to the bisection method for reliability.
t0 = 0.0;
t1 = 1.0;
t2 = x;

if (t2 < t0)
return t0;
if (t2 > t1)
return t1;

while (t0 < t1) {
x2 = sampleCurveX(t2);
if (fabs(x2 - x) < epsilon)
return t2;
if (x > x2)
t0 = t2;
else
t1 = t2;
t2 = (t1 - t0) * .5 + t0;
}

// Failure.
return t2;
}

double solve(double x, double epsilon)
{
return sampleCurveY(solveCurveX(x, epsilon));
}

private:
double ax;
double bx;
double cx;

double ay;
double by;
double cy;
};
}
#endif